HaloRadar: Terminal Video Guide

Quit The Ark and load up The Covenant. Reach the first of the three towers. At one point, you'll jump across a small gap to reach a large elevator platform. All you have to do is continue in the same direction, move across the platform and jump over the opposite gap. You'll end up in a dark room, which brightens considerably when the fourth terminal reveals itself.

Don't worry if you launch the elevator. You can always get the terminal on the way back down.